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376       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

Hence it is enough to show that, for each x e A, the measure mX} x is positive,
or equivalently that f G(x) dmXt x(%) J> 0 for all functions G ^ 0 belonging
to 3fR(Sg) By the same reasoning, we reduce to showing that, for each
y e A, we have f |j>(/)|2 dmx>x(x) *z 0; but by (15.9.2.5)* this is equivalent to
g(y*yx> x) ^ 0, which can also be written g(yx, yx) S> 0 (15.6.3). This com-
pletes the proof of (15.9.2.11).

(15.9.2.12) Every function of the form F =j>, where x, y e A, belongs to <D,
and we have mF = mXt y .

We have to show that, for all u, v in A,

or equivalently (in view of (ii)) that, for all 2 e A,

But the left-hand side of this relation is equal to g(zxy*u, v), and the right-
hand side to g(zuv*x, y). From (15.6.3) and the commutativity of A, we have

g(zxy*u, v) = g(y*zxu, v) = g(zxu, yv)
and

g(zuv*x, y) = g(v*zux, y) = g(zux, vy) = g(zxu, yv).
Hence (15.9.2.12).

We can now prove that every function G 6 tf c(Sg) belongs to $. Let
K = Supp(G); then it is enough to show that there exists a function F e $
which does not vanish on K. For then we can write G = G'F, with G' 6 ^c(S^),
and the lemma (15.9.2.10) will show that G e <D. For each # e K, there exists
by definition an element x e A such that (%) ^ 0, and hence a neighborhood
V(x) of i in Sg such that x(%') ^ 0 for all x' e V(%). Cover K by a finite number
of such neighborhoods V(#j). If xt are the corresponding elements of A, then

the function F =  xtxt does what is required, by virtue of (15.9.2.12).
i

(15.9.2.13)     Definition ofmg and proofof(i).

From (15.9.2.11), the linear form Fi-wF(l) is a positive measure mg on
Sg. Also, for each function F e <J>, we have

(15.9.2.14)                                  JWF = F-HVlinearity of the mapping FH-/%